Download H i - KIAS

Basic Principles of
Quantum computing I
Soonchil Lee
Dept. of physics, KAIST
양자전산의 중요성
• 1 MIPS 컴퓨터로 1016개의 자료 중
하나를 찾을 때
– 고전컴퓨터 : 300 년
– 양자컴퓨터 : 1 분
• 현대 암호는 모두 NSA에서 개발
양자전산 개발을 늦추면 암호종속
모든 정보의 일방적 유출
Classical computing
Quantum computing
i
INPUT
OUTPUT

 H
t
  eiHt /
GATE
OUTPUT U
0
INPUT
Ex) NOT operation
Assign

 eiHt /
0
U
1 i 0
1
  0  
0
0
  1  
1
0 1
We need U  

1 0
For H   H   HI x ,
0
U  exp(i HI xt/ )=exp(itI x )
0 i 1
H
1
Set t  
0 1

U  exp(i I x )  exp(i  x )  i x  i 

2
1 0
고전전산
비트상태
양자전산
전압 0V & 5V 양자고유상태-중첩가능
상태
Ex)spin up & down
Photon olarization
연산자
반도체게이트
알고리듬
수행
게이트의 공간
적 배열을 비
트가 통과
Unitary operation
진화연산자
Optical device
고정된 비트에 연산이 시간
적으로 수행됨
Classical computing
GATE
INPUT
Ex) ADDER
OUTPUT
Quantum computing
GATE
Ex)
U1
INPUT
U2
U3
OUTPUT
U1
U2
t
U3
Execution of quantum algorithm
(1) Algorithm development = a unitary operator U
(2) Decomposition of U : U=U1U2U3… (programming)
where
U i  exp(iH i t / )
and Hi is a part of
H    i I i   J ij I iz I jz
i
i, j
(rf pulse) (J-coupling)
Any unitary operator can be expressed as a sequence
of single qubit operators and controlled-NOT operators.
(3) Real pulse sequence design (compile)
Single qubit operation
|1>
M
dL

dt
 M  H0
  L  H0
H
|0>
   H0
* Single qubit operation
Hi  i Ii
0
1
Ri ( )  exp(iH i t / )
 exp(i ( i I i )t / )
 exp(ii tI i )
 exp(i I i )
Single qubit operation
is done by an rf pulse.
1
(0 1)
2
1
(0 1)
2
1
( 0 i 1 )
2
1
( 0 i 1 )
2
* Controlled-NOT operation
U C  NOT 
Controlled-NOT gate
input
output
C
0
0
1
1
C
0
0
1
1
T
0
1
0
1
T
0
1
1
0




where
Ri ( )  exp(i Ii )
and
U ij ( )  exp(i ( J ij I iz I jz )t / )
 exp(i ( J ij t / ) I iz I jz )
 exp(i I iz I jz )
U ( 0  1 ) 0 : disentangled state
 0 0  1 1 : entangled state

R1z ( ) R2 x ( ) R2 y ( )U12 ( ) R2 y (  )
2
2
2
2
2
Controlled-NOT is done
by just waiting.
Controlled-NOT
input
output
C
0
0
1
1
C
0
0
1
1
T
0
1
0
1
T
0
1
1
0
U ( 0  1 ) 0 : disentangled state
 0 0  1 1 : entangled state
C
T
U
C
T
|11>



|01>
|10>

|00>
Classical computing
f(x)=0
5
4
3
2
3
1
Quantum computing
f(x)=0
|1>+|2>+|3>+….
|3>
Quantum parallel processing
•
Classical parallel processing cannot
imitate because
1. N qubit represents 2N states.
2. entanglement
|1>+|2> = |0>A|1>B+|1>A|0>B
• Shor’s factorization algorithm
– QC : (logN)2+x steps (x<<1)
– classical computer : exp{N1/3(logN)2/3}
– 공개열쇠암호체계 격파
• Grover’s search algorithm
– for N data search, QC : N1/2 try
classical computer : N/2 try
ex) if N=256 & 1 MIPS, 1000 year vs. 4 min.
– 비밀열쇠암호체계 격파
핵자기공명 (NMR: Nuclear Magnetic Resonance)
- 대표적인 핵스핀 조작기법
1) J. Kim, J.-S. Lee, and S. Lee, Phys. Rev. A 61,
032312 (2000).
2) J. Kim, J.-S. Lee, S. Lee, and C. Cheong,
submitted to Phys. Rev. A
Requirements for a Quantum Computer
(1) qubit :
(2) Set :
two quantum states with good quantum #
by measurement or thermal equilibrium
ex) 
(3) Read
(4) Single qubit operation (addressible):
( H   S  U  exp(iHt / )  exp( tS ))
physical addressing or resonance tech.
(5) Interaction (controllable) :
well defined and on-off
( H   JSi S j )
ij
------------------------------------------------------------(6) Coherence : isolation from environment (and other qubits)
(7) Scalability
(1) qubit - two states with good quantum #
•energy : el. floating in LHe
•charge : quantum dot
•spin : quantum dot, molecular magnet, ion trap,
NMR, Si-based QC
•photon : optical QC, cavity QED
•cooper pair : superconductor
•fluxoid : superconductor
Requirements for a Quantum Computer
(1) qubit :
(2) Set :
SPIN
by measurement or thermal equilibrium
ex) 
(3) Read
(4) Single qubit operation (addressible):
( H   S  U  exp(iHt / )  exp( tS ))
physical addressing or resonance tech.
(5) Interaction (controllable) :
well defined and on-off
( H   JSi S j )
ij
(6) Coherence : isolation from environment (and other qubits)
(6) Long coherence : Isolate qubits
•in vacuum : ion trap, el. floating in LHe
•by flying : methods using photon,
el’s trapped by SAW or magnetic field
•in molecule : NMR
•in quantum well : quantum dot, superconductor
•inside solid : Si-based QC
Requirements for a Quantum Computer
(1) qubit :
(2) Set :
SPIN
by measurement or thermal equilibrium
ex) 
(3) Read
(4) Single qubit operation (addressible):
(5) Interaction (controllable) :
well defined and on-off
( H   JSi S j )
ij
(6) Coherence : solid state device
Magnetic Resonance Force Microscopy (MRFM)
- Scanning Probe와 공명의 결합
- 단일스핀 감지
Requirements for a Quantum Computer
(1) qubit :
SPIN
(2) Set
(3) Read : Single spin detection
(4) Single qubit operation (addressible):
(5) Interaction control
( H   JSi S j )
ij
(6) Coherence : solid state device
Ion trap
Qubit - ion spin state
Single spin operation - laser
Inertaction - vibration(CM motion)
Environment
EM
field measurement
Basic Principles of Quantum
computing II
Soonchil Lee
Dept. of physics, KAIST
10 years ago…
• 1st demonstration of quantum
computing by NMR
For 5 years after then…
• We were excited by new challenge.
• Had a hard time to understand new
concepts.
• Lots of NMR QC papers were
published.
• Realized keys of a practical QC.
• Pedestrians show interests.
• Found that NMR is NOT a future QC.
• NMR QC experiment is needed no
more.
Things change. Now…
• Developing a Practical Quantum Computer
is the key issue.
Experiment
Theory
Quantum systems suggested as QC
Atomic and Molecular
Optical
Ion trap
Cavity QED
NMR
Molecular magnet
N@C60(fullerine)
BEC
Photon
Photonic crystal
Solid State
Quantum dot
Superconductor
Si-based QC
Electron beam
el. floating on liquid He
el. trapped by SAW
el. trapped by magnetic field
Requirements for a Quantum Computer
(1)Qubit :
two quantum states with good quantum #
(2) Read : Detection
(3) Single qubit operation (addressible)
(4) Interaction (controllable) :
well defined and on-off
(5) Coherence : isolation from environment
(6) Scalability
Practical Quantum computer
2007.11
Photon
Quantum dot
Josephson
Ion trap
NMR
Si-base QC
Qubit 0
….
5 …. 10 … 20
…..100
Si-based QC (Kane model)
rf coil
magnet
electrode
insulator
Si
P
Si-based QC (Kane model)
Qubit
Si
P
•
•
•
Qubit : nuclear spin of P
Coherence time at 1.5 K
el. spin ~ 103 S
n. spin ~ 10 h
Silicon technology
Read
Addressing
Interaction
Coherence
Scalability
Si-based QC (Kane model)
rf coil
magnet
Qubit
Read
Addressing ?
Interaction ?
Coherence
H
Scalability
Single qubit operation (addressing)
-hyperfine interaction engineering
rf coil
H
magnet
Htotal = Hext+Hhyp
Use electric field to change Hhyp
Single qubit operation (addressing)
-hyperfine interaction engineering
rf coil
++
P atom
B. Kane, Nature 393, 133 (1998)
Interaction control
- RKKY interaction engineering
electrode
10nm
Australian Work
arXiv:cond-mat/0104569
Our strategy
Single spin detection (SET, MRFM)
Kane Model
Ensemble detection (NMR)
P doped
Silicon
Verification of Kane’s QC
model
• 1st step
– Detection of P NMR signal
• 2nd step
– Hyperfine interaction control by E field
• 3rd step
– RKKY interaction control by E field
1st step of Verification of Kane’s QC
• Detection of P NMR signal - never
done
– Fix fluctuating electron spin by low T
and high H to sharpen spectrum.
H
Htotal = Hext+Hhyp
rf coil
Low H
High T
 ph ( H 0  H H .F )
 ph H 0
High
H
Low T
frequency
Experiment
• P NMR of
Si:P with n ~ 1x1017 /cm3
Temp : 45 mK ~ 3.5 K
Field
3He/4He
: 7.3 Tesla
Dilution Refrigerator
(Low Temperature Physics Lab.
Kyoto Univ. )
No signal yet
He
Hex
He
Hex
E field
Hn
Hex
He
Hex
NMR - Direct
Approach
E0
Hhyp
 ph H 0
E0
 ph ( H 0  H H .F )
 ph H 0
frequency
 ph ( H 0  H H .F )
 ph ( H 0  H H .F )
frequency
Electrical control of NMR
frequency
Alternative Approach ESR
Hhf
Quantum Information
Science
• Developing a practical quantum
computer is the key issue.
• We are on a normal research track
after the initial excitement.
• Development goes with
nanotechnology.
• Eventually we will get it!
The
END
2nd step of Verification of Kane’s QC
• Detection of frequency shift by E field
– hyperfine interaction control
rf coil
H
3rd step of Verification of Kane’s QC
• Spectral shape change by electric field
– RKKY interaction control
rf coil
ENDOR
- Sample concentration < 1×1016/cm3
- Temperature < 4K
- Magnetic field T~3.3KG and frequency~9GHz
 ph (H H .F  H 0 )
E0
 ph ( H H .F  H 0 )
E0
We can check NMR
frequency shift by
ENDOR
RF frequency